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Unified representation of the family of L-functions

Abstract

The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.

AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

1 Introduction

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions N={1,2,}, N 0 ={0,1,2,}=N{0}, Z + ={1,2,3,}, Z ={1,2,}. Also, as usual denotes the set of integers, denotes the set of real number and denotes the set of complex numbers. We assume that ln(z) denotes the principal branch of the multi-valued function ln(z) with the imaginary part (ln(z)) constrained by π<(ln(z))π.

Recently, the first author [1] introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:

g a , b (x;t,k,β):= 2 1 k t k e t x β b e t a b = n = 0 Y n , β (x;k,a,b) t n n ! ,
(1)

where (|t|<2π when β=a; |t|<|blog( β a )| when βa; k N 0 ; βC (|β|<1); a,bC{0}).

For the special values of a, b, k, b and β, the polynomials Y n , β (x;k,a,b) provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

Remark 1.1 If we set k=a=b=1 in (1), we get a special case of the generalized Bernoulli polynomials Y n , β (x,k,1,1), that is, the so-called Apostol-Bernoulli polynomials B n (x,β) generated by

t β e t 1 e x t = n = 0 B n (x,β) t n n !

(cf. [128]).

Remark 1.2 By substituting k+1=a=b=1 in (1), we are led to Apostol-Euler polynomials E n (x,β) which are defined by means of the following generating function:

2 β e t + 1 e x t = n = 0 E n (x,β)

(cf. [128]).

Remark 1.3 Setting k=a=b=1 into (1), we get the Apostol-Genocchi polynomials G n (x,β) which are defined by means of the following generating function:

2 t β e t + 1 e x t = n = 0 G n (x,β) t n n !

(cf. [128]).

In terms of a Dirichlet character χ of conductor fN, Ozden et al. [16] extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [[16], p.2783])

Let χ be a Dirichlet character of conductor fN. Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers Y n , χ , β (k,a,b) and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials Y n , χ , β (x;k,a,b) are given by the following generating functions:

F χ , β (t;k,a,b)= 2 1 k t k j = 1 f χ ( j ) ( β a ) b j e j t β b f e f t a b f = n = 0 Y n , χ , β (k,a,b) t n n ! ,
(2)

where (|t|<2π when β=a; |t|<|blog( β a )| when βa; k N 0 ; βC (|β|<1); a,bC{0}) and

H χ , β (x,t;k,a,b)= F χ , β (t,k;a,b) e t x = n = 0 Y n , χ , β (x;k,a,b) t n n !
(3)

(|t|<2π when β=a; |t|<|blog( β a )| when βa; k N 0 ; βC (|β|<1); a,bC{0}).

Remark 1.5 Substituting k=a=b=β=1 into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:

j = 1 f χ ( j ) t e j t e f t 1 = n = 0 B n , χ t n n !
(4)

(cf. [126]).

2 Unification of the L-functions

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials Y n , χ , β (x;k,a,b) in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate Y n , χ , β (x;k,a,b) for negative integer values of n.

Throughout this section, we assume that βC with |β|<1 and sC.

By substituting (1) into (2), we obtain the following functional equation:

F χ , β (t;k,a,b)= 1 f k j = 1 f χ(j) ( β a ) b j g a f , b ( j f , t f ; k , β f ) .
(5)

By using this functional equation, we arrive at the following theorem.

Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have

Y n , χ , β (k,a,b)= f n k j = 1 f χ(j) ( β a ) b j Y n , β f ( j f ; k , a f , b ) .
(6)

By using (5), we modify (3) as follows:

H χ , β (x,t;k,a,b)= 1 f k j = 1 f χ(j) ( β a ) b j g a f , b ( j + x f , t f ; k , β f ) .
(7)

By using (7), we derive the following result.

Corollary 2.2 Let χ be a Dirichlet character of conductor fN. Then we have

Y n , χ , β (x;k,a,b)= f n k j = 1 f χ(j) ( β a ) b j Y n , β f ( j + x f ; k , a f , b ) .
(8)

By applying the Mellin transformation to the generating function (1), Ozden et al. [[16], p.2784 Equation (4.1)] gave an integral representation of the unified zeta function ζ β (s,x;k,a,b):

ζ β (s,x;k,a,b)= 1 Γ ( s ) 0 t s k 1 g a , b (x;t;k,β)dt ( min { ( s ) , ( x ) } > 0 ) ,
(9)

where the additional constraint (x)>0 is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [[16], p.2784 Equation (4.1)] defined the unified zeta function ζ β (s,x;k,a,b) as follows:

ζ β (s,x;k,a,b)= ( 1 2 ) k 1 m = 0 β b m a b ( m + 1 ) ( m + x ) s ( β C ( | β | < 1 ) ; s C ( ( s ) > 1 ) ) .
(10)

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions L χ , β (s,x;k,a,b):

(11)

in terms of the generating function H χ , β (x,t;k;a,b) defined in (7). By substituting (9) into ( 11), we obtain

L χ , β (s,x;k,a,b)= 1 f k + s j = 1 f χ(j) ( β a ) b j ζ β f ( s , j + x f ; k , a f , b )
(12)

where (βC (|β|<1); sC ((s)>1)).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions L χ , β (s,x;k,a,b) as follows.

Definition 2.3 Let χ be a Dirichlet character of conductor fN. For s,βC (|β|<1), we define a two-variable unified L-function L χ , β (s,x;k,a,b) by

L χ , β (s,x;k,a,b)= f k ( 1 2 ) k 1 m = 0 β b m χ ( m ) a b ( m + f ) ( m + x ) s ( β C ( | β | < 1 ) ; ( s ) > 1 ) .
(13)

Remark 2.4 If we substitute x=1 into (13), we get the unified L-function

L χ , β (s;k,a,b):= L χ , β (s,1;k,a,b)

by

L χ , β (s;k,a,b)= f k ( 1 2 ) k 1 m = 1 β b m χ ( m ) a b ( m + f ) m s ,

where ((s)>1, βC (|β|<1)).

Remark 2.5 Upon substituting k=a=b=1 and β= ξ u into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:

l 1 ( u ξ , s , χ ) = L χ , ξ u (s,x;1,1,1),

where, for a positive integer r, ξ is the r th root of 1.

l 1 ( u ξ , s ; χ ) = m = 0 ( ξ u ) m χ ( m ) ( m + x ) s

(cf. [18]).

Remark 2.6 Substituting x=1 into (13), we get a unification of the L-functions

L χ , β (s,1;k,a,b)= L χ , β (s;k,a,b).

Substituting χ1 into (13), we get a unification ζ β (s,x;k,a,b) of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function

ζ(s,x)= ζ 1 (s,x;1,1,1)= n = 0 1 ( n + x ) s

(cf. [27, 28]) and the Riemann zeta function

ζ(s)= ζ 1 (s,1;1,1,1)= n = 1 1 n s

are obvious special cases of the unified zeta function ζ β (s,x;k,a,b) (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function Φ(z,s,a) was given by Ozden et al. [16]:

ζ β (s,x;k,a,b):= ( 1 2 ) k 1 a b Φ ( β b a b , s , x ) ,
(14)

where the Hurwitz-Lerch zeta function is defined by

Φ(z,s,x)= n = 0 z n ( n + x ) s ,

which converges for (x C Z 0 , sC when |z|<1; (s)>1 when |z|=1), where as usual

Z 0 = Z {0}

(cf. [27, 28]).

A relationship between the functions L χ , β (s,x;k,a,b) and ζ β (s,x;k,a,b) is provided by the next theorem.

Theorem 2.7 Let sC. Let χ be a Dirichlet character of conductor fN. Then we have

L χ , β (s,x;k,a,b)= f s k j = 1 f ( β a ) j b χ(j) ζ β f ( s , j + x f ; k , a f , b ) .
(15)

Proof Substituting m=nf+j, j=1,2,,f, n=0,, into (13), we obtain

L χ , β (s,x;k,a,b)= ( 1 2 ) k 1 f s k j = 1 f ( β a ) j b χ(j) n = 0 β b n f a b n f ( n + j + x f ) s .

After some algebraic manipulations, we arrive at the desired result. □

Remark 2.8 Substituting a=b=k=1 into (13), we have

L χ , β (s,x;1,1,1)= m = 0 β m χ ( m ) ( m + x ) s ( ( s ) > 1 , β C ( | β | < 1 ) )

which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:

j = 1 f χ ( j ) t β j e t ( j + x ) β f e t f 1 = n = 0 B n , χ (x,β) t n n ! .

Let f be an odd integer. If we set a=1 and k=0 into (13), then we have

L χ , β (s,x;1,1,1)=2 m = 1 ( 1 ) m χ ( m ) β m ( m + x ) s ( ( s ) > 1 , β C ( | β | < 1 ) ) ,

which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:

j = 1 f 2 χ ( j ) β j e t ( j + x ) β f e t f + 1 = n = 0 E n , χ (x,β) t n n !

(cf. [129]).

By using (15) and (14), we arrive at the following result.

Corollary 2.9 Let sC. Let χ be a Dirichlet character of conductor fN. Then we have

L χ , β (s,x;k,a,b)= ( 1 2 ) k 1 a f b f s k j = 1 f ( β a ) j b χ(j)Φ ( β f b a f b , s , j + x f ) .

Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have

L χ , β (1n,x;k,a,b)= ( 1 ) k f ( n 1 ) ! ( n + k 1 ) ! Y n + k 1 , χ , β (x;k,a,b).
(16)

Proof By substituting s=1n into (15), we get

L χ , β (1n,x;k,a,b)= f n 1 k j = 1 f ( β a ) j b χ(j) ζ β f ( 1 n , j + x f ; k , a f , b ) .

By using Theorem 7 in [16], we get

By substituting (8) into the above, we arrive at the desired result. □

Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions L χ , β (s,x;k,a,b) defined by (13). We thus have (cf. [13])

L(s,x;χ)= m = 0 χ ( m ) ( m + x ) s

and

L(s;χ)= m = 1 χ ( m ) m s ,

where (s)>1. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have

L(1n;χ)= B n , χ n ,

where n Z + and B n , χ , the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that L(s;χ) is non-zero at s=1. Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at s=1 (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

3 Applications

In this section, by using (16) and the following formula, which was proved by Ozden et al. [[16], Theorem 5, Equation (3.10)]

Y n , χ , β (x;k,a,b)= j = 0 n ( n j ) x n j Y j , χ , β (k,a,b),
(17)

we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),

L χ , β (1n,x;k,a,b)= x n + k 1 f l = 0 k 1 ( n + l ) j = 0 n + k 1 ( n + k 1 j ) 1 x j Y j + k 1 , χ , β (k,a,b).

From the above equation, we arrive at the following theorem.

Theorem 3.1 Let x0. Let χ be a Dirichlet character of conductor f. Then we have

L χ , β (s,x;k,a,b)= x k s f l = 0 k 1 ( s 1 l ) j = 0 ( k s j ) 1 x j Y j + k 1 , χ , β (k,a,b).

The function L χ , β (s,x;k,a,b) is an analytic function at s=0. We now compute the value of this function at this point as follows:

L χ , β (0,x;k,a,b)= x k ( 1 ) k f l = 0 k 1 ( 1 + l ) j = 0 k ( k j ) 1 x j Y j + k 1 , χ , β (k,a,b).

The function L χ , β (s,x;k,a,b) is a meromorphic function. This function has simple poles which are

s=1,2,3,,k.

The residues of this function at the simple poles at s=1 and s=k are given, respectively, as follows:

Res s = 1 { L χ , β ( s , x ; k , a , b ) } = x k 1 f ( 1 ) k l = 0 k 1 ( 2 + l ) j = 0 k 1 ( k 1 j ) 1 x j Y j + k 1 , χ , β (k,a,b)

and

Res s = k { L χ , β ( s , x ; k , a , b ) } = Y k 1 , χ , β ( k , a , b ) f l = 0 k 2 ( k 1 l ) .

Remark 3.2 Simsek (cf. [20, 21]) defined a twisted two-variable L-function L ξ , q ( h ) (s,x;χ) as follows:

L ξ , q ( h ) (s,x;χ)= m = 0 χ ( m ) ϕ ξ ( m ) q h m ( x + m ) s log q h s 1 m = 0 χ ( m ) ϕ ξ ( m ) q h m ( x + m ) s 1 ,

where qC (|q|<1); ξ r =1 (rZ); ξ1. Observe that if ξ=1, then L ξ , q ( h ) (s,x;χ) is reduced to the work of Kim [9].

Relationship between the function L χ , β (s,x;k,a,b) and L ξ , q ( h ) (s,x;χ) is given as the following result.

Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have

L 1 , β b a b ( b ) (s,x;χ)= ( 2 ) k a b f f k ( L χ , β ( s , x ; k , a , b ) log q h s 1 L χ , β ( s 1 , x ; k , a , b ) ) .

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function Φ(z,s,a), Hurwitz zeta function ζ(s,x) and L-function (cf. [130]); see also the references cited in each of these earlier works.

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Acknowledgements

Dedicated to Professor Hari M Srivastava.

Both authors are partially supported by Research Project Offices Akdeniz Universities and the Commission of Scientific Research Projects of Uludag University Project number UAP(F) 2011/38 and 2012/16. We would like to thank referees for their valuable comments.

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Ozden, H., Simsek, Y. Unified representation of the family of L-functions. J Inequal Appl 2013, 64 (2013). https://doi.org/10.1186/1029-242X-2013-64

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Keywords

  • Bernoulli numbers
  • Bernoulli polynomials
  • Euler numbers
  • Euler polynomials
  • Genocchi numbers
  • Genocchi polynomials
  • Dirichlet L-functions
  • Hurwitz zeta function
  • Riemann zeta function